| L(s) = 1 | + (0.707 + 2.91i)3-s + (2.82 − 4.12i)5-s + 5.83i·7-s + (−8 + 4.12i)9-s + 16.4i·11-s + (14.0 + 5.33i)15-s − 11.3·17-s − 12·19-s + (−17 + 4.12i)21-s + 24.0·23-s + (−8.99 − 23.3i)25-s + (−17.6 − 20.4i)27-s − 32·31-s + (−48.0 + 11.6i)33-s + (24.0 + 16.4i)35-s + ⋯ |
| L(s) = 1 | + (0.235 + 0.971i)3-s + (0.565 − 0.824i)5-s + 0.832i·7-s + (−0.888 + 0.458i)9-s + 1.49i·11-s + (0.934 + 0.355i)15-s − 0.665·17-s − 0.631·19-s + (−0.809 + 0.196i)21-s + 1.04·23-s + (−0.359 − 0.932i)25-s + (−0.654 − 0.755i)27-s − 1.03·31-s + (−1.45 + 0.353i)33-s + (0.686 + 0.471i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.366172980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366172980\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 2.91i)T \) |
| 5 | \( 1 + (-2.82 + 4.12i)T \) |
| good | 7 | \( 1 - 5.83iT - 49T^{2} \) |
| 11 | \( 1 - 16.4iT - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 11.3T + 289T^{2} \) |
| 19 | \( 1 + 12T + 361T^{2} \) |
| 23 | \( 1 - 24.0T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 32T + 961T^{2} \) |
| 37 | \( 1 + 23.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 57.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 67.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 16.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.83iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 116. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 72T + 6.24e3T^{2} \) |
| 83 | \( 1 - 43.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 65.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 163. iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.923557177378903351105916333879, −9.357977850345361128189613222922, −8.837798285059309039865870483873, −7.959143261563712748793983983926, −6.68621352364781538013835923336, −5.62724639792694116414499484683, −4.85900159307682177157122305753, −4.25090601516816472707600166180, −2.74468650846199785112861535435, −1.80891865629144276471455015704,
0.39137262895045615287089525918, 1.74981777896287236795870128056, 2.91761842898661892359527028051, 3.72120696590037094290479789560, 5.35126320412097729247210610714, 6.28690161872008711440234834490, 6.85339590304184776785341049610, 7.63351942840211747208665899255, 8.615698437230383607780324545303, 9.276412342515670138996764150470
